Rate of Convergence to the Poisson Law of the Numbers of Cycles in the Generalized Random Graphs

Convergence of order O(1/√(n)) is obtained for the distance in totalvariation between the Poisson distribution and the distribution of the numberof fixed size cycles in generalized random graphs with random vertex weights.The weights are assumed to be independent identically distributed randomvariables which have a power-law distribution. The proof is based on theChen–Stein approach and on the derived properties of the ratio of the sum ofsquares of random variables and the sum of these variables. These propertiescan be applied to other asymptotic problems related to generalized randomgraphs.